**NATIONAL OPEN UNIVERSITY OF NIGERIA **

**SCHOOL OF SCIENCE AND TECHNOLOGY **

**COURSE **

**CODE: CIT344 **

**COURSE TITLE: **

ii

**CIT344 **

**INTRODUCTION TO COMPUTER DESIGN **

## Course Team

## Adaora Obayi (Developer/Writer) - NOUN

## Dr. Oyebanji (Programme Leader) - NOUN

## Vivian Nwaocha (Coordinator) -NOUN

**NATIONAL OPEN UNIVERSITY OF NIGERIA **

**COURSE **

iii

## National Open University of Nigeria

## Headquarters

## 14/16 Ahmadu Bello Way

## Victoria Island

## Lagos

## Abuja Office

## 5, Dar es Salaam Street

## Off Aminu Kano Crescent

## Wuse II, Abuja

## Nigeria

## e-mail:

## [email protected]

## URL:

## www.nou.edu.ng

## Published By:

## National Open University of Nigeria

## First Printed 2012

## ISBN: 978-058-047-6

## All Rights Reserved

iv

**CONTENTS **

** PAGE **

## Introduction………. 1

## What You Will Learn in This Course……….………... 1

## Course Aim………. 1

## Course Objectives………... 2

## Working through This Course………...………. 2

## Course Materials……….……… 2

## Study Units……….…………. 3

## Textbooks and References……….……….. 4

## Assignment File……….…………. 4

## Presentation Schedule……….……… 4

## Assessment………. 4

## Tutor-Marked Assignments (TMAs)………....…………..……. 4

## Final Examination and Grading……….……. 4

## Course Marking Scheme……….… 7

## Course Overview……….… 7

## How to Get the Most from This Course………...………….. 8

**Introduction **

**CIT344: Introduction to Computer Design **

## is a 3-credit unit course

## for students studying towards acquiring the Bachelor of Science in

## Information Technology and related disciplines.

## The course is divided into 6 modules and 21 study units. It introduces

## you to concepts in Computer Design and their implementations in our

## everyday lives.

## This course also provides information on numbers and codes in

## computer

## design,

## different

## logic

## designs,

## memory

## devices,

## microprocessors and finally, a type of programming called Assembly

## Language Programming.

## At the end of this course, it is expected that you should be able to

## understand, explain and be adequately equipped with comprehensive

## knowledge of logic designs and can try your hands in some designs of

## your own.

## This course guide therefore gives you an overview of what the course is

## all about, the textbooks and other course materials to be used, what you

## are expected to know in each unit, and how to work through the course

## material.

## Furthermore, it suggests the general strategy to be adopted and also

## emphasises the need for self-assessment and tutor-marked assignment.

## There are also tutorial classes that are linked to this course and you are

## advised to attend them.

**What You Will Learn in This Course **

## The overall aim of this course is to boost your knowledge of logic

## designs, microprocessors and assembly language programming. In the

## course of your studies, you will be equipped with definitions of common

## terms, characteristics and applications of logic designs. You will also

## learn number systems and codes, memory devices, microprocessors and

## finally, loops and subroutines in assembly language.

**Course Aim **

## This course aims to give you an in-depth understanding of computer

## designs. It is hoped that the knowledge would enhance your expertise in

## logic designs.

ii

**Course Objectives **

## It is relevant to note that each unit has its precise objectives. You should

## learn them carefully before proceeding to subsequent units. Therefore, it

## is useful to refer to these objectives in the course of your study of the

## unit to assess your progress. You should always look at the unit

## objectives after completing a unit. In this way, you can be sure that you

## have done what is required of you by the end of the unit. However,

## below are overall objectives of this course. On successful completion of

## this course, you should be able to:

##

## explain the term number system and its various types

##

## state the various conversion from one number system to the other

##

## explain the various types of codes

##

## analyse and design a combinational logic circuit

##

## describe what a sequential logic circuit is

##

## state the differences between combinational and sequential logic

## circuit

##

## list the types of sequential logic circuit

##

## describe what a latch and flip-flop is

##

## describe what shift register is

##

## discuss about finite state machines

##

## describe memory and the basic operations performed on it

##

## state the types of memory we have

##

## describe microprocessors

##

## write a program using assembly language.

**Working through This Course **

## To complete this course, you are required to study all the units, the

## recommended text books, and other relevant materials. Each unit

## contains tutor-marked assignments, and at some point in this course, you

## are required to submit the tutor-marked assignments. There is also a

## final examination at the end of this course. Stated below are the

## components of this course and what you have to do.

**Course Materials **

## The major components of the course are:

## 1.

## Course Guide

## 2.

## Study Units

## 3.

## Text Books

## 4.

## Assignment File

## 5.

## Presentation Schedule

iii

**Study Units **

## There are 6 modules and 21 study units in this course. They are:

**Module 1 **

**Introduction to Numbers and Codes **

## Unit 1

## Types of Number Systems I

## Unit 2

## Types of Number Systems II

## Unit 3

## Codes

**Module 2 **

**Combinational Logic Design and Application **

Unit 1 Analysis and Design of a Combinational Logic Circuit Unit 2 Typical Combinational Logic Circuit I

Unit 3 Typical Combinational Logic Circuit II Unit 4 Typical Combinational Logic Circuit III

**Module 3 **

**Sequential Logic Design and Applications **

## Unit 1

## Sequential Logic Circuits

## Unit 2

## Latches and Flip-Flops

## Unit 3

## Registers

## Unit 4

## Finite State Machines

**Module 4 **

**Memory Devices **

## Unit 1

## Memory Organisation

## Unit 2

## Memory Types

## Unit 3

## Memory Expansion

## Unit 4

## Memory Summary

**Module 5 **

**Introduction to Microprocessors **

## Unit 1

## Microprocessors

## Unit 2

## Central Processing Unit and Arithmetic and Logical Unit

## Unit 3

## Addressing Mode

**Module 6 **

**Assembly Language Programming **

## Unit 1

Learning to Program with Assembly Language## Unit 2

## Branching Loops and Subroutines

iv

**Textbooks and References **

## These texts listed below will be of enormous benefit to you in learning

## this course:

## Baase, Sara (1983).

*VAX-11 Assembly Language Programming*

## . San

## Diego: Prentice-Hall, Inc.

## Brookshear, J. G. (1997).

*Computer Science: An Overview*

## . Addison -

## Wesley.

## Comer, J. David (1994).

*Digital Logic & State Machine Design*

## . Oxford:

## Oxford University Press.

## Dandamudi, S. (2003).

*Fundamentals of Computer Organization and *

*Design*

## . Springer.

## Emerson, W. Pugh

*et al*

## (1991).

*IBM's 360 and Early 370 Systems*

## . MIT

## Press.

## Ford, William & Topp, William (1996).

*Assembly Language and *

*Systems Programming for the M68000 Family*

## . Jones & Bartlett

## Pub.

## Ford, William & Topp, William (1992).

*Macintosh Assembly System*

## ,

*Version 2.0.*

## Jones & Bartlett Pub.

## Hayes, P. John (1993).

*Introduction to Digital Logic*

*Design*

## . Prentice

## Hall.

## Hennessy, L. John & Patterson, A. David (2008).

*Computer *

*Organization & Design*

## . Morgan Kaufmann.

## Holdsworth, Brian & Woods, Clive (2002).

*Digital Logic Design*

## .

## Newnes.

## Hummel, Robert L. (1992).

*Programmer’s Technical Reference: The *

*Processor and Coprocessor*

## . Ziff-Davis Press.

## Mano, M. Morris & Kime, Charles R. (2007).

*Logic and Computer *

*Design Fundamentals*

## . Prentice Hall.

## Marcovitz, Alan (2009).

*Introduction to Logic Design*

## . McGraw-Hill.

## Neamen, Donald (2009).

*Microelectronics: Circuit Analysis & Design*

## .

v

## Nelson, P.

*et al*

## (1995).

*Digital Logic Circuit Analysis & Design*

## .

## Prentice Hall.

## Palmer, James & Periman, David (1993).

*Schaum’s Outline of *

*Introduction to Digital Systems*

## . McGraw-Hill.

## Patterson, David & Hennessy, John (1993).

*Computer Organization & *

*Design: The Hardware/Software Interface*

## . Morgan Kaufmann

## Publishers.

## Patterson, David & Hennessy, John (1990).

*Computer*

*Architecture, A *

*Quantitative Approach*

## . Morgan Kaufmann Publishers.

## Rafiquzzaman, M. (2005).

*Fundamentals of Digital Logic & *

*Microcomputer Design*

## . Wiley – Interscience.

## Roth, H. Charles Jr & Kinney, L. Larry (2009).

*Fundamentals of Logic *

*Design*

## .

## Sanchez, Julio & Maria, P. Canton (1990).

*IBM Microcomputers: A *

*Programmer’s Handbook*

## . McGraw-Hill.

## Struble, George W. (1975).

*Assembler Language Programming The *

*IBM System/360 and 370. *

## Addison-Wesley Publishing Company.

## Tanenbaum, Andrew S. (1998).

*Structured Computer Organization. *

## Prentice Hall. Tocci, J. Ronald & Widmer, S. Neal (1994).

*Digital Systems: Principles and Applications. *

## Tokheium, Roger (1994).

*Schaum’s Outline of Digital Principles*

## .

## McGraw-Hill.

## Wakerly, F. John (2005).

*Digital Design: Principles & Practices *

*Package*

## . Prentice Hall.

## Wear, Larry

* et al. *

## (1991).

*Computers: An Introduction to Hardware and *

*Software Design*

## . McGraw-Hill.

## www.cs.siu.edu

## www.educypedia.be/electronics

## www.books.google.com

vi

**Assignment File **

## The assignment file will be given to you in due course. In this file, you

## will find all the details of the work you must submit to your tutor for

## marking. The marks you obtain for these assignments will count towards

## the final mark for the course. Altogether, there are 21 tutor-marked

## assignments for this course.

**Presentation Schedule **

## The presentation schedule included in this course guide provides you

## with important dates for completion of each tutor-marked assignment.

## You should therefore endeavour to meet the deadlines.

**Assessment **

## There are two aspects to the assessment of this course. First, there are

## tutor-marked assignments; and second, the written examination.

## You are expected to take note of the facts, information and problem

## solving gathered during the course. The tutor-marked assignments must

## be submitted to your tutor for formal assessment, in accordance to the

## deadline given. The work submitted will count for 40% of your total

## course mark. At the end of the course, you will need to sit for a final

## written examination. This examination will account for 60% of your

## total score.

**Tutor-Marked Assignments (TMAs) **

## There are 21 TMAs in this course. You need to submit all the TMAs.

## When you have completed each assignment, send them to your tutor as

## soon as possible and make certain that it gets to your tutor on or before

## the stipulated deadline. If for any reason you cannot complete your

## assignment on time, contact your tutor before the assignment is due to

## discuss the possibility of extension. Extension will not be granted after

## the deadline, unless in extraordinary cases.

**Final Examination and Grading **

## The final examination for CIT344 will be of last for a period of 3 hours

## and have a value of 60% of the total course grade. The examination will

## consist of questions which reflect the self-assessment exercise and

## tutor-marked assignments that you have previously encountered. Furthermore,

## all areas of the course will be examined. It would be better to use the

## time between finishing the last unit and sitting for the examination, to

## revise the entire course. You might find it useful to review your TMAs

vii

## and comment on them before the examination. The final examination

## covers information from all parts of the course.

**Course Marking Scheme **

## The following table includes the course marking scheme

**Table 1: **

**Table 1:**

## Course Marking Scheme

**Assessment **

**Marks **

## Assignments 1-21

## 21 assignments, 40% for the best 4

## Total = 10% X 4 = 40%

## Final Examination

## 60% of overall course marks

## Total

## 100% of Course Marks

**Course Overview **

## This indicates the units, the number of weeks required to complete them

## and the assignments.

**Table 2: **

**Table 2:**

## Course Organiser

**Unit Title of Work **

**Weeks **

**Activity**

**Assessment **

**(End of Unit) **

## Course Guide

## Week 1

** Module 1 Introduction to Numbers and Codes **

## 1

## Types of Number Systems I

## Week 1

## Assignment 1

## 2

## Types of Number Systems II

## Week 2

## Assignment 2

## 3

## Codes

## Week 3

## Assignment 3

** Module 2 Combinational Logic Design and Applications **

## 1

## Analysis & Design of a

## Combinational Logic Circuit

## Week 3

## Assignment 4

## 2

## Typical

## Combinational

## Logic Circuit I

## Week 4

## Assignment 5

## 3

## Typical Combinational Logic

## Circuit II

## Week 4

## Assignment 6

## 4

## Typical

## Combinational

## Logic Circuit III

## Week 5

## Assignment 7

** Module 3 Sequential Logic Design and Applications **

## 1

## Sequential Logic Circuits

## Week 5

## Assignment 8

## 2

## Latches and Flip-Flops

## Week 6

## Assignment 9

## 3

## Registers

## Week 6

## Assignment 10

viii

** Module 4 Memory Devices **

## 1

## Memory Organisation

## Week 7

## Assignment 12

## 2

## Memory Types

## Week 8

## Assignment 13

## 3

## Memory Expansion

## Week 9

## Assignment 14

## 4

## Memory Summary

## Week 10

## Assignment 15

** Module 5 Introduction To Microprocessors **

## 1

## Microprocessors

## Week 10 Assignment 16

## 2

## Central Processing Unit &

## Arithmetic & Logical Unit

## Week 11

## Assignment 17

## 3

## Addressing Mode

## Week 12

## Assignment 18

** Module 6 Assembly Language Programming **

## Unit

## 1

## Learning to Program with

## Assembly Language

## Week 13

## Assignment 19

## Unit

## 2

## Branching

## Loops

## and

## Subroutine

## Week 14

## Assignment 20

## Unit

## 3

## Sample

## Programs

## in

## Assembly Language

## Week 14

## Assignment 21

**How to Get the Most Out of This Course **

## In distance learning, the study units replace the university lecturer. This

## is one of the huge advantages of distance learning mode; you can read

## and work through specially designed study materials at your own pace

## and at a time and place that is most convenient. Think of it as reading

## from the teacher, the study guide indicates what you ought to study, how

## to study it and the relevant texts to consult. You are provided with

## exercises at appropriate points, just as a lecturer might give you an

## in-class exercise.

## Each of the study units follows a common format. The first item is an

## introduction to the subject matter of the unit and how a particular unit is

## integrated with the other units and the course as a whole. Next to this is

## a set of learning objectives. These learning objectives are meant to guide

## your studies. The moment a unit is finished, you must go back and

## check whether you have achieved the objectives. If this is made a habit,

## then you will increase your chances of passing the course.

## The main body of the units also guides you through the required

## readings from other sources. This will usually be either from a set book

## or from other sources. Self assessment exercises are provided

## throughout the unit, to aid personal studies and answers are provided at

## the end of the unit. Working through these self tests will help you to

## achieve the objectives of the unit and also prepare you for tutor marked

## assignments and examinations. You should attempt each self test as you

## encounter them in the units.

ix

## Read the course guide thoroughly and organise a study schedule. Refer

## to the course overview for more details. Note the time you are expected

## to spend on each unit and how the assignment relates to the units.

## Important details, e.g. details of your tutorials and the date of the first

## day of the semester are available. You need to gather together all these

## information in one place such as a diary, a wall chart calendar or an

## organiser. Whatever method you choose, you should decide on and write

## in your own dates for working on each unit.

## Once you have created your own study schedule, do everything you can

## to stick to it. The major reason that students fail is that they get behind

## with their course works. If you get into difficulties with your schedule,

## please let your tutor know before it is too late for help.

## Turn to unit 1 and read the introduction and the objectives for the unit.

## Assemble the study materials. Information about what you need for a

## unit is given in the table of content at the beginning of each unit. You

## will almost always need both the study unit you are working on and one

## of the materials recommended for further readings, on your desk at the

## same time.

## Work through the unit, the content of the unit itself has been arranged to

## provide a sequence for you to follow. As you work through the unit, you

## will be encouraged to read from your set books.

## Keep in mind that you will learn a lot by doing all your assignments

## carefully. They have been designed to help you meet the objectives of

## the course and will help you pass the examination.

## Review the objectives of each study unit to confirm that you have

## achieved them. If you are not certain about any of the objectives, review

## the study material and consult your tutor.

## When you are confident that you have achieved a unit’s objectives, you

## can start on the next unit. Proceed unit by unit through the course and try

## to pace your study so that you can keep yourself on schedule.

## When you have submitted an assignment to your tutor for marking, do

## not wait for its return before starting on the next unit. Keep to your

## schedule. Pay particular attention to your tutor’s comments on the

## tutor-marked assignment form and also written on the assignment when the

## assignment is returned to you. Consult you tutor as soon as possible if

## you have any questions or problems.

x

## After completing the last unit, review the course and prepare yourself

## for the final examination. Check that you have achieved the unit

## objectives (listed at the beginning of each unit) and the course objectives

## (listed in this course guide).

**Facilitators/Tutors and Tutorials **

## There are 8 hours of tutorial provided in support of this course. You will

## be notified of the dates, time and location together with the name and

## phone number of your tutor as soon as you are allocated a tutorial group.

## Your tutor will mark and comment on your assignments, keep a close

## watch on your progress and on any difficulties you might encounter and

## provide assistance to you during the course. You must mail your tutor

## marked assignment to your tutor well before the due date. At least two

## working days are required for this purpose. They will be marked by your

## tutor and returned to you as soon as possible. Do not hesitate to contact

## your tutor by telephone, e-mail or discussion board if you need help.

## The following might be circumstances in which you would find help

## necessary:

##

## you do not understand any part of the study units or the assigned

## readings

##

## you have difficulty with the self test or exercise

##

## you have questions or problems with an assignment, with your

## tutor’s comments on an assignment or with the grading of an

## assignment.

## You should try your best to attend the tutorials. This is the only chance

## to have face-to-face contact with your tutor and ask questions which are

## answered instantly. You can raise any problem encountered in the

## course of your study. To gain the maximum benefit from the course

## tutorials, prepare a question list before attending them. You will learn a

## lot from participating actively in tutorial discussions.

xi

## Course Code

## CIT344

## Course Title

## Introduction to Computer Design

## Course Team

## Adaora Obayi (Developer/Writer) - NOUN

## Dr. Oyebanji (Programme Leader) - NOUN

## Vivian Nwaocha (Coordinator) -NOUN

xii

## National Open University of Nigeria

## Headquarters

## 14/16 Ahmadu Bello Way

## Victoria Island

## Lagos

## Abuja Office

## 5, Dar es Salaam Street

## Off Aminu Kano Crescent

## Wuse II, Abuja

## Nigeria

## e-mail:

## [email protected]

## URL:

## www.nou.edu.ng

## Published By:

## National Open University of Nigeria

## First Printed 2012

## ISBN: 978-058-047-6

## All Rights Reserved

xiii

**CONTENTS **

** PAGE **

**Module 1 **

**Introduction to Numbers and Codes…………..… **

** 1 **

## Unit 1

## Types of Number Systems I……….…..… 1

## Unit 2

## Types of Number Systems II……….. 11

## Unit 3

## Codes……….... 21

**Module 2 **

**Combinational Logic Design and Application .….. 28 **

Unit 1 Analysis and Design of a Combinational Logic Circuit….. 28

Unit 2 Typical Combinational Logic Circuit I……….…… 31

Unit 3 Typical Combinational Logic Circuit II……….... 39

Unit 4 Typical Combinational Logic Circuit III……….. 51

**Module 3 **

**Sequential Logic Design and Applications…….… 60 **

## Unit 1

## Sequential Logic Circuits………..….. 60

## Unit 2

## Latches and Flip-Flops……….……..

## 65

## Unit 3

## Registers……….………

## 90

## Unit 4

## Finite State Machines………..………… 105

**Module 4 **

**Memory Devices………..…… 126 **

## Unit 1

## Memory Organisation……… 126

## Unit 2

## Memory Types………... 135

## Unit 3

## Memory Expansion……… 150

## Unit 4

## Memory Summary……….. 154

**Module 5 **

**Introduction to Microprocessors………. 157 **

## Unit 1

## Microprocessors……….. 157

## Unit 2

## Central Processing Unit and Arithmetic and Logical

## Unit………. 165

## Unit 3

## Addressing Mode……… 175

**Module 6 **

**Assembly Language Programming………... 188 **

## Unit 1

Learning to Program with Assembly Language…………... 188## Unit 2

## Branching Loops and Subroutines

…………...## 205

1

**MODULE 1 **

**INTRODUCTION TO NUMBERS AND **

**CODES **

## Unit 1

## Types of Number Systems I

## Unit 2

## Types of Number Systems II

## Unit 3

## Codes

**UNIT 1 TYPES OF NUMBER SYSTEMS I **

**CONTENTS **

## 1.0

## Introduction

## 2.0

## Objectives

## 3.0

## Main Content

## 3.1

## Decimal Number System

## 3.2

## Binary Number System

## 3.2.1

## Fractions in Binary Number System

## 3.2.2

## Binary Arithmetic

## 3.2.3

## Binary to Decimal Conversion

## 3.2.4

## Decimal to Binary Conversion

## 4.0

## Conclusion

## 5.0

## Summary

## 6.0

## Tutor-Marked Assignment

## 7.0

## References/Further Reading

**1.0**

**INTRODUCTION **

## The number system is the basis of computing. It is a very important

## foundation for understanding the way the computer system works. In

## this unit, we will talk about decimal and binary number system.

## Endeavour to assimilate as much as possible from this unit – especially,

## the conversion from one number system to another.

**2.0**

**OBJECTIVES **

## At the end of this unit, you should be able to:

##

## explain the term decimal number system

##

## manipulate fractions of decimal numbers

##

## explain the term binary number system

##

## manipulate binary arithmetic

##

## convert binary to decimal

2

**3.0 **

**MAIN CONTENT **

**3.1 **

**Decimal Number System **

## The decimal number system has ten unique digits: 0, 1, 2, 3, 4, 5, 6, 7, 8,

## 9. Using these single digits, ten different values can be represented.

## Values greater than ten can be represented by using the same digits in

## different combinations. Thus, ten is represented by the number 10, two

## hundred seventy five is represented by 275, etc. Thus, same set of

## numbers 0, 1, 2, … 9 are repeated in a specific order to represent larger

## numbers. The decimal number system is a positional number system as

## the position of a digit represents its true magnitude. For example, 2 is

## less than 7, however 2 in 275 represents 200, whereas 7 represents 70.

## The left most digit has the highest weight and the right most digit has

## the lowest weight. 275 can be written in the form of an expression in

## terms of the base value of the number system and weights.

## 2 x 102 + 7 x 101 + 5 x 100 = 200 + 70 + 5 = 275

## where, 10 represents the base or radix,102, 101, 100 represent the

## weights 100, 10 and 1 of the numbers 2, 7 and 5.

**Fractions in Decimal Number System **

## In a Decimal Number System the fraction part is separated from the

## integer part by a decimal point. The integer part of a number is written

## on the left hand side of the decimal point. The fraction part is written on

## the right side of the decimal point. The digits of the integer part on the

## left hand side of the decimal point have weights 100, 101, 102 etc.

## respectively starting from the digit to the immediate left of the decimal

## point and moving away from the decimal point towards the most

## significant digit on the left hand side. Fractions in decimal number

## system are also represented in terms of the base value of the number

## system and weights. The weights of the fraction part are represented by

## 10-1, 10-2, 10-3, etc. The weights decrease by a factor of 10 moving

## right of the decimal point. The number 382.91 in terms of the base

## number and weights is represented as

## 3 x 102 + 8 x 101 + 2 x 100 + 9 x 10

-1## + 1 x 10

-2## = 300 + 80 + 2 + 0.9 +

## 0.01 = 382.91

3

**3.2 **

**Binary Number System **

## Binary as the name indicates is a base-2 number system having only two

## numbers 0 and 1. The binary digit 0 or 1 is known as a ‘Bit’. Below is

## the decimal equivalent of the binary number system.

**Table 1: **

**Table 1:**

## Decimal Equivalents of Binary Number System

**Decimal **

**Number **

**Binary **

**Number **

**Decimal **

**Number **

**Binary **

**Number **

## 0

## 0

## 10

## 1010

## 1

## 1

## 11

## 1011

## 2

## 10

## 12

## 1100

## 3

## 11

## 13

## 1101

## 4

## 100

## 14

## 1110

## 5

## 101

## 15

## 1111

## 6

## 110

## 16

## 10000

## 7

## 111

## 17

## 10001

## 8

## 1000

## 18

## 10010

## 9

## 1001

## 19

## 10011

## 20

## 10100

## Counting in binary number system is similar to counting in decimal

## number systems. In a decimal number system a value larger than 9 has

## to be represented by 2, 3, 4, or more digits. Similarly, in the binary

## number system a binary number larger than 1 has to be represented by 2,

## 3, 4, or more binary digits.

## Any binary number comprising of binary 0 and 1 can be easily

## represented in terms of its decimal equivalent by writing the binary

## number in the form of an expression using the base value 2 and weights

## 20, 21, 22, etc.

## The number 10011

2## (the subscript 2 indicates that the number is a binary

## number and not a decimal number ten thousand and eleven) can be

## rewritten in terms of the expression:

## 10011

2## = (1 x 24) + (0 x 23) + (0 x 22) + (1 x 21) + (1 x 20)

## = (1 x 16) + (0 x 8) + (0 x 4) + (1 x 2) + (1 x 1)

## = 16 + 0 + 0 + 2 + 1

4

**3.2.1 Fractions in Binary Number System **

## In a decimal number system the integer part and the fraction part of a

## number are separated by a decimal point. In a binary number system the

## integer part and the Fraction part of a binary number can be similarly

## represented separated by a decimal point. The binary number 1011.101

2## has an integer part represented by 1011 and a fraction part 101 separated

## by a decimal point. The subscript 2 indicates that the number is a binary

## number and not a decimal number. The binary number 1011.101

2## can be

## written in terms of an expression using the base value 2 and weights 23,

## 22, 21, 20, 2

-1## , 2

-2## and 2

-3## .

## 1011.101

2## = (1 x 23) + (0 x 22) + (1 x 21) + (1 x 20) + (1 x 2

-1## ) + (0 x 2

-2## ) + (1 x 2

-3## )

## = (1 x 8) + (0 x 4) + (1 x 2) + (1 x 1) + (1 x 1/2) + (0 x 1/4) + (1 x 1/8)

## = 8 + 0 + 2 + 1 + 0.5 + 0 + 0.125

## = 11.625

## Computers do handle numbers such as 11.625 that have an integer part

## and a fraction part. However, it does not use the binary representation

## 1011.101. Such numbers are represented and used in floating-point

## numbers notation.

**3.2.2 Binary Arithmetic **

## Digital systems use the binary number system to represent numbers.

## Therefore these systems should be capable of performing standard

## arithmetic operations on binary numbers.

**Binary Addition **

## Binary addition is identical to decimal addition. By adding two binary

## bits, a sum bit and a carry bit are generated. The only difference between

## the two additions is the range of numbers used. In binary addition, four

## possibilities exist when two single bits are added together. The four

## possible input combinations of two single bit binary numbers and their

## corresponding sum and carry outputs are specified in Table 2.

**Table 2: **

**Table 2:**

## Addition of Two Single Bit Binary Numbers

## First Number

## Second Number Sum

## Carry

## 0

## 0

## 0

## 0

## 0

## 1

## 1

## 0

## 1

## 0

## 1

## 0

5

## The first three additions give a result 0, 1 and 1 respectively which can

## be represented by a single binary digit (bit). The fourth addition results

## in the number 2, which can be represented in binary as 102. Thus, two

## digits (bits) are required. This is similar to the addition of 9 + 3 in

## decimal. The answer is 12 which cannot be represented by a single digit;

## thus, two digits are required. The number 2 is the sum part and 1 is the

## carry part.

## Any number of binary numbers having any number of digits can be

## added together.

**Binary Subtraction **

## Binary subtraction is identical to decimal subtraction. The only

## difference between the two is the range of numbers. Subtracting two

## single bit binary numbers results in a difference bit and a borrow bit.

## The four possible input combinations of two single bit binary numbers

## and their corresponding difference and borrow outputs are specified in

## Table 3. It is assumed that the second number is subtracted from the first

## number.

**Table 3: **

**Table 3:**

** Subtraction of Two Single Bit Binary Numbers **

## First Number

## Second Number Difference

## Borrow

## 0

## 0

## 0

## 0

## 0

## 1

## 1

## 1

## 1

## 0

## 1

## 0

## 1

## 1

## 0

## 0

## The second subtraction subtracts 1 from 0 for which a borrow is

## required to make the first digit equal to 2. The difference is 1. This is

## similar to decimal subtraction when 17 is subtracted from 21. The first

## digit 7 cannot be subtracted from 1, therefore 10 is borrowed from the

## next significant digit. Borrowing a 10 allows subtraction of 7 from 11

## resulting in a difference of 4.

**Binary Multiplication **

## Binary multiplication is similar to the decimal multiplication except for

## the range of numbers. Four possible combinations of two single bit

## binary numbers and their products are listed in table 4.

6

**Table 4: **

**Table 4:**

## Multiplication of two Single Bit Binary Numbers

## First Number

## Second Number

## Product

## 0

## 0

## 0

## 0

## 1

## 0

## 1

## 0

## 0

## 1

## 1

## 1

**Binary Division **

## Division in binary follows the same procedure as in the division of

## decimal numbers. Fig 1 illustrates the division of binary numbers.

## 10

## 101 | 1101

## 101

## 011

## 000

## 11

**Fig. 1 : Binary Division **

**3.2.3**

**Binary to Decimal Conversion **

## Most real world quantities are represented in decimal number system.

## Digital systems on the other hand are based on the binary number

## system. Therefore, when converting from the digital domain to the

## real-world, binary numbers have to be represented in terms of their decimal

## equivalents. The method used to convert from binary to decimal is the

## sum-of-weights method.

**Sum-of-Weights Method **

## Sum-of-weights as the name indicates sums the weights of the binary

## digits (bits) of a binary number which is to be represented in decimal.

## The sum-of-weights method can be used to convert a binary number of

## any magnitude to its equivalent decimal representation.

## In the sum-of-weights method an extended expression is written in terms

## of the binary base number 2 and the weights of the binary number to be

## converted. The weights correspond to each of the binary bits which are

## multiplied by the corresponding binary value.

## Binary bits having the value 0 do not contribute any value towards the

## final sum expression. The binary number 10110

2## is therefore written in

7

## corresponding to the bits 0, 1, 1, 0 and 1 respectively. Weights 2

0## and 2

3## do not contribute in the final sum as the binary bits corresponding to

## these weights have the value 0.

## 10110

2## = 1 x 2

4## + 0 x 2

3## + 1 x 2

2## + 1 x 2

1## + 0 x 2

0## = 16 + 0 + 4 + 2 + 0

## = 22

**Sum-of-Non-Zero Terms **

## In the sum-of-weights method, the binary bits 0 do not contribute

## towards the final sum representing the decimal equivalent. Secondly, the

## weight of each binary bit increases by a factor of 2 starting with a

## weight of 1 for the least significant bit. For example, the binary number

## 101102 has weights 2

0## =1, 2

1## =2, 2

2## =4, 2

3## =8 and 2

4## =16 corresponding to

## the bits 0, 1, 1, 0 and 1 respectively.

## The sum-of-non-zero terms method is a quicker method to determine

## decimal equivalents of binary numbers without resorting to writing an

## expression. In the sum-of-zero terms method, the weights of

## non-zero binary bits are summed, as the weights of non-zero binary bits do not

## contribute towards the final sum representing the decimal equivalent.

## The weights of binary bits starting from the right most least significant

## bit is 1, The next significant bit on the left has the weight 2, followed by

## 4, 8, 16, 32, etc. corresponding to higher significant bits. In binary

## number system the weights of successive bits increase by an order of 2

## towards the left side and decrease by an order of 2 towards the right

## side. Thus, a binary number can be quickly converted into its decimal

## equivalent by adding weights of non-zero terms which increase by a

## factor of 2. Binary numbers having an integer and a fraction part can

## similarly be converted into their decimal equivalents by applying the

## same method.

## A quicker method is to add the weights of non-zero terms. Thus, for the

## numbers:

## 10011

2## = 16 + 2 + 1 = 19

## 1011.101

2## = 8 + 2 + 1 + ½ + 1/8 = 11 + 5/8 = 11.625

**3.2.4**

**Decimal to Binary Conversion **

## Conversion from decimal to binary number system is also essential to

## represent real-world quantities in terms of binary values. The

## sum-of-weights and repeated division by 2 methods are used to convert a

## decimal number to equivalent binary.

8

**Sum-of-Weights **

## The sum-of-weights method used to convert binary numbers into their

## decimal equivalent is based on adding binary weights of the binary

## number bits. Converting back from the decimal number to the original

## binary number requires finding the highest weight included in the sum

## representing the decimal equivalent. A binary 1 is marked to represent

## the bit which contributed its weight in the sum representing the decimal

## equivalent. The weight is subtracted from the sum decimal equivalent.

## The next highest weight included in the sum term is found. A binary 1 is

## marked to represent the bit which contributed its weight in the sum term

## and the weight is subtracted from the sum term. This process is repeated

## until the sum term becomes equal to zero. The binary 1s and 0s

## represent the binary bits that contributed their weight and bits that did

## not contribute any weight respectively.

## The process of determining binary equivalent of a decimal number 392

## and 411 is illustrated in a tabular form.

**Table 5: **

**Table 5:**

## Converting Decimal to Binary using Sum-of-Weights

## Method

## Sum Term

## Highest

## Weight

## Binary Number Sum Term

## = Sum Term – Highest

## Weight

## 392

## 256

## 100000000 136

## 136

## 128

## 110000000 8

## 8

## 8

## 110001000

## 0

## The sum-of-weights method requires mental arithmetic and is a quick

## way of converting small decimal numbers into binary. With practice

## large decimal numbers can be converted into binary equivalents.

**Repeated Division-by-2 **

## Repeated division-by-2 method allows decimal numbers of any

## magnitude to be converted into binary. In this method, the decimal

## number to be converted into its binary equivalent is repeatedly divided

## by 2. The divisor is selected as 2 because the decimal number is being

## converted into binary a base-2 number system. Repeated division

## method can be used to convert decimal number into any number system

## by repeated division by the base-number.

## In the repeated-division method the decimal number to be converted is

## divided by the base number, in this particular case 2. A quotient value

## and a remainder value is generated, both values are noted done. The

9

## remainder value in all subsequent divisions would be either a 0 or a 1.

## The quotient value obtained as a result of division by 2 is divided again

## by 2. The new quotient and remainder values are again noted down. In

## each step of the repeated division method the remainder values are noted

## down and the quotient values are repeatedly divided by the base number.

## The process of repeated division stops when the quotient value becomes

## zero. The remainders that have been noted in consecutive steps are

## written out to indicate the binary equivalent of the original decimal

## number.

**Table 6: **

**Table 6:**

## Converting Decimal to Binary using Repeated

## Division by 2 Method

## Number Quotient after division Remainder after division

## 392 196

## 0

## 196

## 98

## 0

## 98

## 49

## 0

## 49

## 24

## 1

## 24

## 12

## 0

## 12

## 6

## 0

## 6

## 3

## 0

## 3

## 1

## 1

## 1

## 0

## 1

## The process of determining the binary equivalent of a decimal number

## 392 is illustrated in a tabular form above. Reading the numbers in the

## remainder column from bottom to top 110001000 gives the binary

## equivalent of the decimal number 392.

**SELF-ASSESSMENT EXERCISE **

## Explain with the aid of good examples, the different methods of

## converting binary numbers to decimal numbers.

**4.0**

**CONCLUSION **

## The decimal number system has ten unique digits 0, 1, 2, 3… 9. Using

## these single digits, ten different values can be represented. Values

## greater than ten can be represented by using the same digits in different

## combinations. Binary indicates a base-2 number system having only two

## numbers 0 and 1. The binary digit 0 or 1 is known as a ‘Bit’.

**5.0**

**SUMMARY**

## In this unit, we discussed decimal and binary number systems,

## manipulation of their fractions, binary arithmetic and conversion of

10

## decimal to binary and vice versa. Hoping that you understood the topics

## discussed, you may now attempt the questions below.

**6.0**

**TUTOR-MARKED ASSIGNMENT **

## 1.

## Briefly explain these terms: decimal number system, binary

## number system.

## 2.

## Convert these decimal numbers to binary

## (a)

## 105 (b) 345 (c) 55

## 3.

## Convert these binary numbers to decimal

## (a)

## 10110.101 (b) 1111.111 (c) 110100100

**7.0**

**REFERENCES/FURTHER READING **

## Comer, J. David (1994).

*Digital Logic & State Machine Design*

## . Oxford

## University Press.

## Dandamudi, S. (2003).

*Fundamentals of Computer Organization and *

*Design*

## . Springer.

## Hayes, P. John (1993).

*Introduction to Digital Logic*

*Design*

## . Prentice

## Hall.

## Hennessy, L. John & Patterson, A. David (2008).

*Computer *

*Organization & Design*

## . Morgan Kaufmann.

## Holdsworth, Brian & Woods, Clive (2002).

*Digital Logic Design*

## .

## Newnes.

## Mano, M. Morris & Kime, Charles R. (2007).

*Logic and Computer *

*Design Fundamentals*

## . Prentice Hall. Pg 283.

## Marcovitz Alan (2009).

*Introduction to Logic Design*

## . McGraw-Hill.

## Nelson, P Victor,

*et al*

## (1995).

*Digital Logic Circuit Analysis & Design*

## .

## Prentice Hall.

## Palmer James & Periman David (1993).

*Schaum’s Outline of *

*Introduction to Digital Systems*

## . McGraw-Hill.

## Rafiquzzaman M. (2005).

*Fundamentals of Digital Logic & *

*Microcomputer Design*

## . Wiley – Interscience.

## www.cs.siu.edu

## www.educypedia.be/electronics

## www.books.google.com

11

**UNIT 2 TYPES OF NUMBER SYSTEMS II **

**CONTENTS **

## 1.0

## Introduction

## 2.0

## Objectives

## 3.0

## Main Content

## 3.1 Hexadecimal Number System

## 3.1.1 Counting in Hexadecimal Number System

## 3.1.2 Binary to Hexadecimal Conversion

## 3.1.3 Hexadecimal to Binary Conversion

## 3.1.4 Decimal to Hexadecimal Conversion

## 3.1.5 Hexadecimal to Decimal Conversion

## 3.1.6 Hexadecimal Addition and Subtraction

## 3.2

## Octal Number System

## 3.2.1

## Counting in Octal Number System

## 3.2.2

## Binary to Octal Conversion

## 3.2.3

## Octal to Binary Conversion

## 3.2.4

## Decimal to Octal Conversion

## 3.2.5 Octal to Decimal Conversion

## 3.2.6 Octal Addition and Subtraction

## 4.0

## Conclusion

## 5.0

## Summary

## 6.0

## Tutor-Marked Assignment

## 7.0

## References/Further Reading

**1.0**

**INTRODUCTION **

## In this unit we shall conclude with hexadecimal and octal number

## systems.

**2.0**

**OBJECTIVES **

## At the end of this unit, you should be able to:

##

## explain the term hexadecimal number system

##

## count in hexadecimal

##

## convert binary to hexadecimal

##

## convert hexadecimal to binary

##

## convert decimal to hexadecimal

##

## convert hexadecimal to decimal

##

## explain hexadecimal addition and subtraction

##

## explain the term octal number system

12

**3.0 **

**MAIN CONTENT **

**3.1**

**Hexadecimal Numbe**

## r

** System **

## Representing even small number such as 6918 requires a long binary

## string (1101100000110) of 0s and 1s. Larger decimal numbers would

## require lengthier binary strings. Writing such long string is tedious and

## prone to errors.

## The hexadecimal number system is a base 16 number system and

## therefore has 16 digits and is used primarily to represent binary strings

## in a compact manner. Hexadecimal number system is not used by a

## digital system. The hexadecimal number system is for our convenience

## to write binary strings in a short and concise form. Each hexadecimal

## number digit can represent a 4-bit binary number. The binary numbers

## and the hexadecimal equivalents are listed below:

**Table 1: **

**Table 1:**

## Hexadecimal Equivalents of Decimal and Binary Numbers

## Decimal Binary Hexadecimal Decimal Binary Hexadecimal

## 0 0000 0 8 1000 8

## 1 0001 1 9 1001 9

## 2 0010 2 10 1010 A

## 3 0011 3 11 1011 B

## 4 0100 4 12 1100 C

## 5 0101 5 13 1101 D

## 6 0110 6 14 1110 E

## 7 0111 7 15 1111 F

**3.1.1 Counting in Hexadecimal**

**Number System **

## Counting in hexadecimal is similar to the other number systems already

## discussed. The maximum value represented by a single hexadecimal

## digit is F which is equivalent to decimal 15. The next higher value

## decimal 16 is represented by a combination of two hexadecimal digits

## 10

16## or 10 H. The subscript 16 indicates that the number is hexadecimal

## 10 and not decimal 10. Hexadecimal numbers are also identified by

## appending the character H after the number. The hexadecimal numbers

## for decimal numbers 16 to 39 are listed below in table 2:

13

**Table 2: **

**Table 2:**

## Counting using Hexadecimal Numbers

Decimal Hexadecimal Decimal Hexadecimal Decimal Hexadecimal

## 16

## 10

## 24

## 18

## 32

## 20

## 17

## 11

## 25

## 19

## 33

## 21

## 18

## 12

## 26

## 1A

## 34

## 22

## 19

## 13

## 27

## 1B

## 35

## 23

## 20

## 14

## 28

## 1C

## 36

## 24

## 21

## 15

## 29

## 1D

## 37

## 25

## 22

## 16

## 30

## 1E

## 38

## 26

## 23

## 17

## 31

## 1F

## 39

## 27

**3.1.2 Binary to Hexadecimal Conversion **

## Converting binary to hexadecimal is a very simple operation. The binary

## string is divided into small groups of 4-bits starting from the least

## significant bit. Each 4-bit binary group is replaced by its hexadecimal

## equivalent.

## 11010110101110010110 binary number 1101 0110 1011 1001 0110

## Dividing into groups of 4-bits

## D 6 B 9 6 Replacing each group by its hexadecimal

## equivalent

## Thus, 11010110101110010110 is represented in hexadecimal by D6B96

## Binary strings which cannot be exactly divided into a whole number of

## 4-bit groups are assumed to have 0’s appended in the most significant

## bits to complete a group.

## 1101100000110 Binary Number

## 1 1011 0000 0110 Dividing into groups of 4-bits

## 0001 1011 0000 0110 Appending three 0s to complete the group

## 1 B 0 6 Replacing each group by its hexadecimal

## equivalent

**3.1.3 Hexadecimal to Binary Conversion **

## Converting from Hexadecimal back to binary is also very simple. Each

## digit of the hexadecimal number is replaced by an equivalent binary

## string of 4-bits.

14

## F D 1 3 hexadecimal number

## 1111 1101 0001 0011 Replacing each hexadecimal digit by its 4-bit

## binary equivalent.

**3.1.4 Decimal to Hexadecimal Conversion **

## There are two methods to convert from decimal to hexadecimal. The

## first method is the indirect method and the second method is the

## repeated division method.

**Indirect Method **

## A decimal number can be converted into its hexadecimal equivalent

## indirectly by first converting the decimal number into its binary

## equivalent and then converting the binary to Hexadecimal.

**Repeated Division-by-16 Method **

## The repeated division method has been discussed earlier and used to

## convert decimal numbers to binary by repeatedly dividing the decimal

## number by 2. A decimal number can be directly converted into

## hexadecimal by using repeated division. The decimal number is

## continuously divided by 16 (base value of the hexadecimal number

## system).

## The conversion of decimal 2096 to hexadecimal using the repeated

## division-by-16 method is illustrated in Table 3. The hexadecimal

## equivalent of 2096

10## is 830

16## .

**Table 3: **

**Table 3:**

## Hexadecimal Equivalent of Decimal Numbers Using

## Repeated Division

## Number Quotient after division Remainder after division

## 2096 131 0

## 131 8 3

## 8 0 8

**3.1.5 Hexadecimal to Decimal Conversion **

## Converting hexadecimal numbers to decimal is done using two methods.

## The first method is the indirect method and the second method is the

## sum-of-weights method.

15

**Indirect Method **

## The indirect method of converting hexadecimal number to decimal

## number is to first convert hexadecimal number to binary and then binary

## to decimal.

**Sum-of-Weights Method **

## A hexadecimal number can be directly converted into decimal by using

## the sum of weights method. The conversion steps using the

## sum-of-weights method are shown.

## CA02 hexadecimal number

## C x 16

3## + A x 16

2## + 0 x 16

1## + 2 x 16

0## Writing the number in an

## expression

## (C x 4096) + (A x 256) + (0 x 16) + (2 x 1)

## (12 x 4096) + (10 x 256) + (0 x 16) + (2 x 1) Replacing hexadecimal

## values with

## decimal equivalents

## 49152 + 2560 + 0 + 2 Summing the weights

## 51714 Decimal equivalent

**3.1.6 Hexadecimal Addition and Subtraction **

## Numbers represented in hexadecimal can be added and subtracted

## directly without having to convert them into decimal or binary

## equivalents. The rules of addition and subtraction that are used to add

## and subtract numbers in decimal or binary number systems apply to

## hexadecimal addition and subtraction. Hexadecimal addition and

## subtractions allows large binary numbers to be quickly added and

## subtracted.

**Hexadecimal Addition **

## Carry 1

## Number 1 2 AC 6

## Number 2 9 2 B 5

## Sum B D 7 B

**Hexadecimal Subtraction **

## Borrow 1 1 1

## Number 1 9 2 B 5

## Number 2 2 A C 6

## Difference 6 7 E F

16

**3.2 **

**Octal Number System **

## Octal number system also provides a convenient way to represent long

## string of binary numbers. The octal number is a base 8 number system

## with digits ranging from 0 to 7. Octal number system was prevalent in

## earlier digital systems and is not used in modern digital systems

## especially when the hexadecimal number is available. Each octal

## number digit can represent a 3-bit binary number. The binary numbers

## and the octal equivalents are listed below

**Table 4: **

**Table 4:**

## Octal Equivalents of Decimal and Binary Numbers

## Decimal Binary Octal

## 0 000

## 0

## 1 001 1

## 2 010 2

## 3 011 3

## 4 100 4

## 5 101 5

## 6 110 6

## 7 111 7

**3.2.5**

**Counting in Octal Number System **

## Counting in octal is similar to counting in any other number system. The

## maximum value represented by a single octal digit is 7. For representing

## larger values a combination of two or more octal digits has to be used.

## Thus, decimal 8 is represented by a combination of10

8## . The subscript 8

## indicates the number is octal 10 and not decimal ten. The octal numbers

## for decimal numbers 8 to 30 are listed below:

**Table 5: **

**Table 5:**

## Counting using Octal Numbers

## Decimal Octal Decimal Octal Decimal Octal

## 8 10 16 20 24 30

## 9 11 17 21 25 31

## 10 12 18 22 26 32

## 11 13 19 23 27 33

## 12 14 20 24 28 34

## 13 15 21 25 29 35

## 14 16 22 26 30 36

## 15 17 23 27 31 37

17

**3.2.6**

**Binary to Octal Conversion **

## Converting binary to octal is a very simple. The binary string is divided

## into small groups of bits starting from the least significant bit. Each

## 3-bit binary group is replaced by its octal equivalent.

## 111010110101110010110 Binary number

## 111 010 110 101 110 010 110 Dividing into groups of 3-bits

## 7 2 6 5 6 2 6 Replacing each group by its octal

## equivalent

## Thus, 111010110101110010110 is represented in octal by 7265626

## Binary strings which cannot be exactly divided into a whole number of

## 3-bit groups are assumed to have 0’s appended in the most significant

## bits to complete a group.

## 1101100000110 Binary number

## 1 101 100 000 110 Dividing into groups of 3-bits

## 001 101 100 000 110 Appending three 0s to complete the group

## 1 5 4 0 6 Replacing each group by its octal equivalent

**3.2.7**

**Octal to Binary Conversion **

## Converting from octal back to binary is also very simple. Each digit of

## the octal number is replaced by an equivalent binary string of 3-bits.

## 1 7 2 6 Octal number

## 001 111 010 110 Replacing each octal digit by its 3-bit binary

## equivalent.

**3.2.8**

**Decimal to Octal Conversion **

## There are two methods to convert from decimal to octal. The first

## method is the

## Indirect Method and the second method is the repeated division method.

**Indirect Method **

## A decimal number can be converted into its octal equivalent indirectly

## by first converting the decimal number into its binary equivalent and

## then converting the binary to octal.

18

**Repeated Division-by-8 Method **

## The repeated division method has been discussed earlier and used to

## convert decimal numbers to binary and hexadecimal by repeatedly

## dividing the decimal number by 2 and 16 respectively. A decimal

## number can be directly converted into octal by using repeated division.

## The decimal number is continuously divided by 8 (base value of the

## Octal number system).

## The conversion of decimal 2075 to octal using the repeated

## division-by-8 method is illustrated in Table 6. The octal equivalent of 2075

10## is

## 4033

8## .

**Table 6: **

**Table 6:**

## Octal Equivalent of Decimal Numbers Using Repeated

## Division

## Number Quotient after Division Remainder after Division

## 2075 259 3

## 259 32 3

## 32 4 0

## 4 0 4

**3.2.5 Octal to Decimal Conversion **

## Converting octal numbers to decimal is done using two methods. The

## first method is the indirect method and the second method is the

## sum-of-weights method.

**Indirect Method **

## The indirect method of converting octal number to decimal number is to

## first convert octal number to binary and then binary to decimal.

**Sum-of-Weights Method **

## An octal number can be directly converted into decimal by using the

## sum of weights method. The conversion steps using the sum-of-weights

## method are shown.

## 4033 octal number

## 4 x 8

3## + 0 x 8

2## + 3 x 8

1## + 3 x 8

0## Writing the number in an expression

## (4 x 512) + (0 x 64) + (3 x 8) + (3 x 1)

## 2048 + 0 + 24 + 3 Summing the weights

## 2075 Decimal equivalent

19